%TO DO: in tabel FCTS waarden toevoegen
\section{Results}
In all our results certain values are taken for $\Delta x $ and $\Delta t$
these values have a big influence on the premium therefore the values taken
will be specified each time. The boundaries on the x-axis influence the
premium a great deal as well, therefore the $X_{min}$ and $X_{max}$ will be
specified as well.\\
The model we use calculates these boundaries in the following way:
\begin{eqnarray}\label{eq:xmin}
X_{min}'&=&X_0-\Delta x \cdot X_{min}
\end{eqnarray}
\begin{eqnarray}\label{eq:xmax}
X_{max}'&=&X_0+\Delta x \cdot X_{max} 
\end{eqnarray}
in which $X_0=\log(S_0)$.\\
\\
\noindent
We have used the parameters shown in table \ref{tab:calloption} to calculate
the option price for an European call and compared the results with the
analytical Black-Scholes formulae. From the results in this table it is clear
that the approximation found by both Finite Difference schemes are comparable.
However as will be discussed later in this work, the major advantage of the
Crank-Nicolson scheme over the Forward Time Centered scheme is that the
parameters of the latter are restricted by the stability of the scheme. 

\begin{table}[htbp]
\caption{Approximation of the price of different call options}\label{tab:calloption}
\begin{center}
\begin{tabular}{l*{8}{c}}
\hline
&r&$\sigma$&$S_0$&K&T&Analytic Result&\multicolumn{2}{c}{Approximation}\\
&&&&&&(Black-Scholes)&CNS&
	FCTS\\
\hline\hline
In the money&4\%&30\%&100&110&1&9.6254&9.6203&9.6205\\
At the money&4\%&30\%&110&110&1&15.1286&15.1131&15.1133\\
Out of the money&4\%&30\%&120&110&1&21.7888&21.7766&21.7768\\
\hline
\end{tabular}
\\\medskip
{\footnotesize To compare both approximations boundaries were chosen to be
$\log(S_0) - 0.5$ and $\log(S_0) + 1.0$, $\Delta t = 0.0001$ and $\Delta x =
0.025$. These values for $\Delta t$ and $\Delta x$ were chosen based on the
stability analysis treated later in this work.}
\end{center}
\end{table}

\begin{figure}[htbp]
\begin{center}
\caption{Approximation of the value of a call option given several step sizes}\label{fig:callapp}
\includegraphics[width=15cm]{call_cns}
{\footnotesize The call option which was approximated is at the money for a
strike price of 110.
The volatility is $30 \%$ and the interest rate is $4\%$. The analytical solution gives a
premium of $15.1286$. The borders of the grid used to approximate the premiums
in this figures are fixed on $\log(110) - 0.5$ and $\log(110) + 1$.}
\end{center}
\end{figure}

To analyze the effects the size $\Delta x$ and $\Delta t$ have on the quality
of the approximation the premium was approximated with different finite
Difference grids. The results for a put option is shown in figure
\ref{fig:putapp}. For this figure the Crank-Nicolson scheme is used since its
lack of instability permits more combinations of $\Delta x$ and $\Delta t$.
The figure clearly shows that decreasing the size of $\Delta x$ increases the
result less than increasing the number of discretization steps of the time.
Moreover, the figure also shows that increasing the precision of the Finite
Difference grid does not improve the approximation constantly, the
approximation improves considerably from 0.1 to 0.001 but increases only
faintly from that point onwards. This holds for $\Delta x$ as well as $\Delta
t$.


\begin{figure}[htbp]
\begin{center}
\caption{Approximation of the value of a put option given several step sizes}\label{fig:putapp}
\includegraphics[width=15cm]{premium_approx_cns}
{\footnotesize The put option which was approximated is at the money for a
strike price of 110.
The volatility is $30 \%$ and the interest rate is $4\%$. The analytical solution gives a
premium of $10.8154$. The borders of the grid used to approximate the premiums
in this figures are fixed on $\log(110) - 0.75$ and $\log(110) + 0.5$.}
\end{center}
\end{figure}

In Finite Difference the choice of the grid dimension is the key to correct
premium approximations. As became evident in the previous figures, the choice
of $\Delta t$ and $\Delta x$ is crucial to a good approximation. However,
these are not the only essential parameters. Also the minimum and maximum
value of the stock used in approximation play a major role in the quality of
the approximation. This because they lay at the origin of the quality of the
approximation of the payoff. 

Different choices for the boundary of the stock price lead to better
approximations of the premium. As can be seen in figure \ref{fig:putopt} the
quality of the approximation of the premium of a put option is only good with
$X_{min} = 500$ and $X_{max} = 40$ (the blue line almost overlaps the red
line which shows the analytical solution). For a call option the magnitude of
the factors is the opposite. The left sub figure of figure \ref{fig:callopt}
shows that the analytical result can not be approximated as well as with a put
option. The right figure shows however, that with a $X_{min} = 40$ and
$X_{max} = 60$ the premium is considerably well approximated.

\begin{figure}[htbp]
\begin{center}
\caption{Approximation of the value of a call option given different $X_{min}$ and $X_{max}$}\label{fig:callopt}
\includegraphics[width=6cm]{fdcnscall}
\includegraphics[width=6cm]{fdcnscall_40-65}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\caption{Approximation of the value of a put option given different $X_{min}$ and $X_{max}$}\label{fig:putopt}
\includegraphics[width=6cm]{fdcnsput}
\includegraphics[width=6cm]{fdcnsput_30-65}
\end{center}
\end{figure}

% vim: spell spelllang=en:autoindent

